The central limit theorem for Markov chains started at a point. (English) Zbl 1012.60028

Derriennic, Yves; Lin, Michael

Probab. Theory Relat. Fields 125, No. 1, 73-76 (2003).

Summary: The aim of this paper is to prove a central limit theorem and an invariance principle for an additive functional of an ergodic Markov chain on a general state space, with respect to the law of the chain started at a point. No irreducibility assumption nor mixing conditions are imposed; the only assumption bears on the growth of the \(L^2\)-norms of the ergodic sums for the function generating the additive functional, which must be \(O(n^\alpha)\) with \(\alpha<\frac 12\). The result holds almost surely with respect to the invariant probability of the chain.

MSC:

60F05	Central limit and other weak theorems
60J05	Discrete-time Markov processes on general state spaces

Keywords:

central limit theorem; invariance principle; additive functional; ergodic Markov chain